LEC-4 CL 601 Tensor algebra and its application in continuum mechanics.pptx

Tensor algebra;
continuum mechanics;
Strain tensor;
Stress tensor;
Stress invariants;
deviatoric stress;
Octahedral stresses;
Stiffness;
Geotechnical Engineering;
M Tech Geotechnical Engineering.;
Constitutive modelling

1. Constitutive Modelling of Geomaterials Prof. Samirsinh P Parmar Mail: [email protected] Asst. Prof. Department of Civil Engineering, Faculty of Technology, Dharmasinh Desai University, Nadiad , Gujarat, INDIA Lecture: 4 : Tensor algebra and its application in continuum mechanics

2. 3 . Tensor algebra and its application in continuum mechanics

3. Basic properties of tensor Most of the second order tensor and vector operation follows regular matrix oper ation Addition Transpose P  Q  R P  Q T Scalar product   Q : R   Q   R Product of tensor and vector Product of two tensors   Q ji R ij c  Qs c  Q T s C  QR C  Q T R Trace of tensors tr  P   P 11  P 22  P 33 c i  Q ij s j c i  Q ji s j C ij  Q ik R kj C ij  Q ki R kj tr  P   P ij  ij Not every second order object is a tensor. Only those objects are tensors, whose components transform according to the following transformation law. Transformation of tensors P ij *  Q ik Q jl P kl P ij  Q ij  R ij P ij  Q ji   Q ij R ij

4. Basic properties of tensor

5. Application of tensor in continuum mechanics The property of a tensor -­‐ to associate a vector with any plane in the continuum -­‐ is the major reason why we use tensors in Continuum Mechanics. This property also allows us to define tensor components in mechanical applications  i  Q ij n j Strain tensor The macroscopic geometric changes within a non-­‐rigid body due to external forces are termed as deformation These deformations are described using the strain tensor A finite volume cannot disappear; A finite volume cannot become infinite; Continuous body remains continuous during and after deformation; Lagrangian approach (The motion is described by the material coordinate) Eulerian approach (The motion is described by the spatial coordinate) Finite strain/ large strain

6. Strain tensor Small-­‐strain : linear terms >> non-­‐linear terms Lagrangian strain ≈ Eulerian strain Cauchy’s infinitesimal strain tensor

7. Strain tensor (geometric interpretation) Linear strain Deformation: Engineering linear strain: m= n

8. Strain tensor (geometric interpretation) Shear strain m ≠ n

9. Strain tensor Decomposition of the strain tensor Volumetric strain (change in volume)  v   11   22   33  I 1    ij  ij   kk Deviatoric strain (change in shape) e ij   ij    ij   kk  ij I  1  ij 3 3 1 J  2  2 e ij e ij

10. Stress tensor Body, B = B 1 + B 2 t i   ij n j Traction vector ( N/m 2 ) The trac tion vector will be different for other sections passing through the same point P The tensor which provides the complete information about the traction vector for any arbitrary section passing through the point P is the stress tensor In case of strain tensor , a similar approach can be introduced using stretch vector : s i = ε ij n i

11. Stress tensor t t t Cauchy stress tensor = stress tensor = true stress tensor Normal stress :  n  t i n i   ij n j n i Shear stress: 2  n  t i t i   n   ij n j  ik n k   n 2 In 3D any vector should have 3 components stress tensor must be symmetric due to moment equilibrium

12. If there is a plane exist, where the stress vector is perpendicular to the plane (i.e., shear stress is zero ), it is called a principal plane , and its normal is called a principal direction (or axis ). A coordinate system with axes oriented to the principal directions implies that the normal stresses are the principal stresses and the stress tensor is represented by a diagonal matrix: Principal shear stresses/maximum shear stress 1  max  2   1   3  Principal stresses

13. Stress invariants Characteristics equation of a stress tensor σ ij for the condition  ij   ij  

14. Stress invariants In terms of principle stress: 1 Decomposition of stress tensor  ij  s ij  3  kk  ij  s ij   ij    p Mean stress/ hydrostatic stress (pressure) s i j Deviatoric stress This decomposition is analogous to strain decomposition

15. Invariants of deviatoric stress

16. Octahedral stresses In principal axis the plane is define such a way that the normal stress on it is proportional to the first invariant of the stress tensor I 1 , while the shear stress is proportional to the square root of the second invaria nt of the deviatoric stress tensor J 2 . I 1  o c t  3  oct  2 J 2 3 Generalized measures of the hydrostatic and deviatoric loading of the body at the point

17. Stiffness: Constitutive model parameters are calibrated through laboratory tests which provides the connection between the stress-­‐strain responses of the materials.  ij  D ijkl  kl Stiffness is a fourth order tensor D ij k l The tensor D ijkl has 3 4 =81 components   D  Stiffness tensor holds minor symmetry, so we can write D ijkl = D jikl = D ijlk = D jilk So, number of component will reduced to, 36 Not a matrix!!!!

18. Stiffness: Tensor matrix representation

19. Stiffness: Isotropic linear elastic stress/strain relation D ijk l    i j  k l     i k  j l   i l  j k      i j 1  ij  2   3   2    kk  2   ij  ij   ij  kk  2  ij Generalized Hooks law

20. Stiffness: Generalized Hooks law

21. Numerical problems: A   B  C    A  C  B   A  B  C where A , B and C are three vectors in 3D space Prove the a bove identity using general vector algebra; using index notation and graphical method 2) Determine the first stress invariant ( I 1 ), second & third deviatoric stress invariant ( J 2 & J 3 ) of the following stress tensor; also determine the components of principal stress tensor from the invariants . 1)

22. 3) In a drained triaxial compression test, the stress state of a cylindrical specimen at the verge of failure is the following:  a  600kPa and  r  200kPa Determine the principal stress tensor at this condition. Further compute the first stress invariant and second deviatoric stress invariant. shear stress and normal stress at the potential failure plane considering mobilized friction angle of the material is 30° . Determine the vometric strain on the specimen assuming linear elastic isotropic stiffness, Bulk modulus = 33.33 Mpa and Poisson's ratio = 0.25 Comment on the correlation between stress and strain invariants. Numerical problems:

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